Temperature controlling method, thermal treating apparatus, and method of manufacturing semiconductor device

ABSTRACT

A temperature control method is provided which is capable of performing quick, accurate, and error-free soaking control over all wafer areas to be thermally treated at a target temperature without requiring any skilled operator and which can be automated by using a computer. In the temperature control method of controlling a heating apparatus having at least two heating zones in such a manner that temperatures detected at predetermined locations equal a target temperature therefor, temperatures are detected at predetermined locations the number of which is larger than the number of the heating zones, and the heating apparatus is controlled in such a manner that the target temperature falls between a maximum value and a minimum value of a plurality of detected temperatures.

BACKGROUND OF THE INVENTION

[0001] 1. Field of the Invention

[0002] The present invention relates to a temperature controllingmethod, a thermal treating apparatus for a semiconductor manufacturingdevice or the like, and a method for manufacturing a semiconductordevice, and in particular, it relates to a temperature controllingmethod and a thermal treating apparatus for a semiconductor device orthe like, and a method for manufacturing a semiconductor device, whereinin order to thermally treat a treatment target, the thermal treatingapparatus is divided into a plurality of heating zones, and targettemperatures are set for the plurality of heating zones for temperaturecontrol so that the target temperatures are corrected using temperaturesdetected in areas of the treatment target the number of which is largerthan the number of the plurality of heating zones.

[0003] 2. Description of the Related Art

[0004] With known thermal treating apparatuses, if, a semiconductorwafer (substrate) as a treatment target is thermally treated for filmformation or the like, various temperatures are required whichcorrespond to the types of films formed as a result of the treatment anda fabrication process thereof. Accordingly, during the thermaltreatment, the temperature of the treatment target is controlled so asto be as close to the treatment temperature as possible (soaking controlmethod). FIG. 6 is a diagram showing the structure of a verticaldiffusion furnace, which is typical of such thermal treatmentapparatuses. The vertical diffusion furnace shown in FIG. 6 is composedof a soaking pipe 112 and a reaction pipe 113 covered by an outer wall111, a heater 114 for heating the inside of the reaction pipe 113,heater thermocouples 115 for detecting the temperatures of the heater114 at a plurality of locations thereof, cascade thermocouples 116 fordetecting the temperature at locations between the soaking pipe 112 andthe reaction pipe 113, a boat 117 on which a wafer to be subjected tothermal treatment is mounted, and a temperature controller 119 forcontrolling the amount of operation Z (value of electric power) for theheater 114 on the basis of the temperatures detected by the heaterthermocouple 115 and the cascade thermocouple 116 as well as anindicated target temperature Y.

[0005] The heater 114 is divided into a plurality of heating zones so asto accurately control the in-furnace temperature (temperature of thereaction pipe 113). If the heater 114 is divided, for example, into fourheating zones as shown in FIG. 6, the zones are called U, CU, CL, and Lzones (hereinafter these names will be used) respectively from the topto the bottom of the figure. Each of the heating zones has the heaterthermocouples 115 and the cascade thermocouples 116 installed therein.In order that the temperatures detected by the cascade thermocouples 116equal the target temperature, the temperature controller 119 detects thetemperatures of the heater thermocouples 115 while calculating theamount of operation Z for the heater 114 in accordance with a givenalgorithm (PID calculations or the like), thereby adjusting the powervalue for the heater 114.

[0006] In this manner, the conventional temperature control is executedso that the temperatures detected by the cascade thermocouples 116 equalthe target value for the wafer treatment. Accordingly, there may be nota small difference between the temperature at the location of the waferto be actually treated and the corresponding temperature detected by thecascade thermocouple, thereby degrading the quality of the thermaltreatment. Thus, it is necessary to control the temperature of an areacloser to the wafer or the temperature of the wafer itself so as toequal the target value for the wafer treatment in order to improve thequality of the thermal treatment. To achieve this, a means is requiredwhich detects the temperature of an area closer to the wafer or thetemperature of the wafer itself. The detection means includes variousmethods such as the one of inserting thermocouples into the reactiontube to measure the temperatures of the neighborhoods of the wafer, orestimating the wafer temperatures using a mathematical model. Here, byway of example, a description will be given of a method of usingtemperature measuring wafers (thermocouple-mounted wafers) eachcomprising a thermocouple directly installed on a wafer.

[0007]FIG. 7 shows an example using the above describedthermocouple-mounted wafers. In this case, thermocouple-mounted wafers118 are arranged so as to correspond to the U, CU, CL, and L zones, thefour heating zones. The heater thermocouples 115 and the cascadethermocouples 116 are also installed at locations corresponding to theU, CU, CL, and L zones. The temperatures detected by thethermocouple-mounted wafers 118 are obtained not only by the heaterthermocouples 115 and the cascade thermocouples 116 but also by thetemperature controller 119. Further, for the thermocouple-mountedwafers, the location at which the thermocouple is installed and thenumber of thermocouples installed may depend on the usage thereof. Forthe thermocouple-mounted wafers 118 in the example described herein, itis assumed that only one thermocouple is installed in the center of thewafer.

[0008]FIG. 9 shows an example of the relationship between thetemperatures detected by the cascade thermocouples 116 and thethermocouple-mounted wafers 118, wherein the temperature control isexecuted so that the temperatures detected by the cascade thermocouples116 equal the target value for the wafer treatment. In this case, thetemperatures (∘) detected by the cascade thermocouples 116 equal thetarget value, whereas there may be errors between the temperatures (Δ)detected by the thermocouple-mounted wafers 118 and the target value.Further, since the magnitude of the error and the like varies among theheating zones, this constitutes a factor reducing the quality of thethermal treatment. In this case, the errors between the temperaturesdetected by the thermocouple-mounted wafers 118 and the targettemperature for the cascade thermocouples 116 may be used as correctivevalues for this target temperature. For example, in FIG. 9, if thetemperature detected by the thermocouple-mounted wafer for the U zone islower than the target value by 5° C., then this 5° C. can be used as acorrective value for the target temperature for the correspondingcascade thermocouple.

[0009] The above described correction increases the temperature detectedby the cascade thermocouple 116 for the U zone, above the originaltarget value by 5° C., but the temperature detected by thethermocouple-mounted wafer for the U zone can be made equal to theoriginal target value. FIG. 10 shows an example of the relationshipbetween the temperatures detected by the cascade thermocouples and thethermocouple-mounted wafers, wherein the correction is executed for allthe heating zones. In this case, the temperatures (∘) detected by thecascade thermocouples 116 do not equal the original target value,whereas the temperatures (Δ) detected by the thermocouple-mounted wafers118 equal the target value. The temperature of wafers to be actuallythermally treated equals the target value, so that the quality of thethermal treatment can be improved. In this example, however, even if 5°C. is added to the target value for the cascade thermocouple as acorrective value in order to increase the temperature detected by thethermocouple-mounted wafer 118 by 5° C., the temperature detected by thethermocouple-mounted wafer often fails to actually increase by 5° C.,thereby requiring an adjustment operation to be repeated several times.

[0010] Furthermore, in the construction shown in FIG. 7, thethermocouple-mounted wafers 118 are arranged at the locationscorresponding to the heater thermocouples 115 and cascade thermocouples116 installed for the respective heating zones. In contrast, FIG. 11shows an example of the relationship between the temperatures detectedby the cascade thermocouples and the thermocouple-mounted wafers,wherein in order to measure the temperatures of other wafers, additionalplural thermocouple-mounted wafers are arranged as shown in FIG. 8 andtemperatures detected thereby are similarly obtained by the temperaturecontroller 119. In this case, the temperatures (Δ) detected by thethermocouple-mounted wafers 118 arranged at locations representative ofthe corresponding heating zones equal the target value, whereas thetemperatures (▴) detected by the thermocouple-mounted wafers arranged atlocations different from those mentioned above have errors with respectto the target value. This may lead to differences in the quality of thethermal treatment, thus reducing the rate at which products having aquality of a fixed level are manufactured. To prevent this, the targettemperature for the cascade thermocouples 116 may further be correctedso as to minimize the differences in temperature between the wafer areasto thereby obtain a uniform temperature.

[0011] For example, if the temperature detected by thethermocouple-mounted wafer installed between the CL zone and the L zoneis 3° C. higher than the target value, the target temperature for thecascade thermocouple for the CL and L zones are reduced by about 1° C.as a corrective value. In this case, the corrective value is set at 1°C. instead of 3° C. because if the error of 3° C. is used as acorrective value for the target temperature for the cascade thermocoupleas described above, then with respect to the temperatures detected bythe thermocouple-mounted wafers corresponding to the heating zones theerror is too large compared to the target temperature, therebypreventing the attainment of the object to eliminate the difference intemperature between the wafer areas. Another explanation for the settingof the corrective value at 1° C. is as follows. The temperature detectedby the thermocouple-mounted wafer installed between the CL zone and theL zone is affected by corrections for the CL and L zones. Accordingly,if information such as the level of the interference between the heatingzones is insufficient, the corrective value must be adjusted a number oftimes. Thus, the initial value is set at almost 1° C.

[0012]FIG. 12 shows an example of the relationship between thetemperatures detected by the cascade thermocouples and thethermocouple-mounted wafers, wherein with the construction as shown inFIG. 8, the temperatures detected by all the thermocouple-mounted wafersare adjusted (for example, by a skilled operator) so as to reduce theerrors with respect to the target value. In this case, the temperatures(Δ) detected by the thermocouple-mounted wafers corresponding to theheating zones slightly deviates from the target value, but for all thethermocouple-mounted wafers (Δ, ▴), the errors in the detectedtemperatures with respect to the target value are generally smaller thanin FIG. 11 (the width of the variation shown by the arrow is smaller).This serves to increase the number of products having a quality of afixed level or higher. A skilled operator, however, is required inreducing the errors in the thermocouple-mounted wafers with respect tothe target temperature as shown in FIG. 12. Further, it presently takeseven skilled operators much time to adjust the errors because theadjustment operation must be repeated many times.

[0013] The above described conventional soaking control method forthermal treatment apparatuses comprises dividing the wafer areas of thethermal treatment apparatus into a plurality of heating zones, actuallysetting target temperatures for all the heating zones, detecting thetemperatures of areas subjected to temperature using temperaturedetecting devices such as the cascade thermocouples in peripheries ofthe furnace, and providing heating control using the detectedtemperatures so that treatment targets arranged in the furnace can betreated at the target temperature. Since, however, the determination ofa set temperature for the temperature detecting devices depends on theskilled operator's experience or trials, the number of proper ablepersonnel is limited and much time is required for the setting.

SUMMARY OF THE INVENTION

[0014] The present invention is intended to obviate the above problems,and has for its object to provide a temperature control method, athermal treatment apparatus, and a method of manufacturing asemiconductor device, which can simply and promptly adjust (soakingcontrol) the temperatures of all areas of a treatment target to a targetvalue while reducing resulting errors and which can be easily automatedusing a computer system.

[0015] Bearing the above object in mind, according to a first aspect ofthe present invention, there is provided a temperature control method ofcontrolling a heating apparatus having at least two heating zones so asto adjust temperatures detected at predetermined locations to a targetvalue therefor, the method comprising: detecting temperatures at thepredetermined locations the number of which is larger than the number ofthe heating zones and at least one of which is in each of the heatingzones; and controlling the heating apparatus in such a manner that thetarget temperature falls between a maximum value and a minimum value ofa plurality of temperatures detected at a plurality of detectedpredetermined locations.

[0016] With this configuration, even without any skilled operator, thetemperatures of all areas of a treatment target can be simply andpromptly adjusted (soaking control) to a target value while reducingresulting errors. Thus, the present invention is easily applicable to avertical CVD apparatus or a sheet-feed apparatus which has a pluralityof heating zones and which allows the detection of temperatures at thelocations of thermocouple-mounted wafers.

[0017] In a preferred form of the first aspect of the present invention,first temperature detectors are disposed at first predeterminedlocations corresponding to the respective zones, and are used for atemperature control method of controlling the heating apparatus in sucha manner that temperatures detected by the first temperature detectorsequal a first target temperature. Second temperature detectors aredisposed at second predetermined locations which are closer to atreatment target than the first predetermined locations, to obtain aninterference matrix M as well as differences P₀ between a second targettemperature for the second temperature detectors and temperaturesdetected by the second temperature detectors, the interference matrix Mbeing a matrix of coefficients indicative of the extents of variationsof temperatures detected by the second temperature detectors when thefirst target temperature for the first temperature detectors is varied.The first target temperature is corrected on the basis of theinterference matrix M and the errors P₀.

[0018] With the this configuration, even if there is not any skilledoperator, the temperatures of all areas of the treatment target can besimply and promptly adjusted (soaking control) to a target value whilereducing resulting errors, and the system can be automated using acomputer system. Here, note that in an embodiment of the invention, thefirst temperature detectors correspond to cascade thermocouples, and thesecond temperature detectors correspond to thermocouples attached towafers (thermocouple-mounted wafers). With this construction,temperature control can be carried out while correcting the targettemperature for the cascade thermocouples on the basis of theinterference matrix M and the errors P₀ obtained.

[0019] In another preferred form of the first aspect of the presentinvention, the temperature control method further comprises: determiningnew errors P₀′ by performing temperature control using the correctedfirst target temperature; and correcting the corrected first targettemperature using the new errors P₀′ and the interference matrix M.

[0020] With this configuration, the temperature control can beaccurately carried out to precisely heat the treatment target at adesired temperature.

[0021] According to a second aspect of the present invention, there isprovided a temperature control method for controlling an apparatus whichincludes a process chamber, a heating apparatus having at least oneheating zone for heating a treatment target provided in the processchamber, and first temperature detectors provided at least one for eachzone for detecting heating temperatures provided by the heatingapparatus at first predetermined locations, wherein the heatingapparatus is controlled on the basis of first detected temperaturesdetected by the first temperature detectors and a first targettemperature for the first detected temperatures, and wherein a pluralityof second temperature detectors are disposed at second predeterminedlocations the number of which is larger than that of the heating zonesand which are closer to the treatment target than the firstpredetermined locations, the second temperature detectors being operableto detect heating temperatures provided by the heating apparatus. Themethod comprises: comparing second detected temperatures detected by thesecond temperature detectors with a second target temperature for thesecond detected temperatures to obtain corrective values for the firsttarget temperature; and correcting the first target temperature by thecorrective values to perform temperature control.

[0022] In a preferred form of the second aspect of the presentinvention, the corrective values are obtained before an actual processof actually treating a substrate to be treated.

[0023] With the above configurations of the second aspect of theinvention, no second temperature detector needs to be provided in theactual process, thereby preventing the adverse effects of the provisionof the temperature detectors on the treatment target.

[0024] According to a third aspect of the present invention, there isprovided a thermal treatment apparatus comprising: a process chamber; aheating apparatus having at least two heating zones and being subjectedto temperature control in such a manner that temperatures detected atpredetermined locations equal a target temperature therefor; a pluralityof temperature detectors for detecting temperatures at predeterminedlocations the number of which is larger than the number of the heatingzones and at least one of which is in each of the heating zones; and acontrol device for controlling the heating apparatus in such a mannerthat the target temperature falls between a maximum value and a minimumvalue of a plurality of temperatures detected by means of the pluralityof temperature detectors.

[0025] With this configuration, a thermal treatment apparatus can beprovided with which even without any skilled operator, the temperaturesof all areas of a treatment target can be simply and promptly adjusted(soaking control) to a target value while reducing resulting errors.

[0026] According to a fourth aspect of the present invention, there isprovided a method of manufacturing a semiconductor device, in which atarget substrate is subjected to a heating process by controlling aheating apparatus having at least two heating zones in such a mannerthat temperatures detected at predetermined locations equal a targettemperature therefor, the method comprising: detecting temperatures atpredetermined locations the number of which is larger than the number ofthe heating zones and at least one of which is in each of the heatingzones; and controlling the heating apparatus in such a manner that thetarget temperature falls between a maximum value and a minimum value ofa plurality of temperatures detected at a plurality of detectedpredetermined locations.

[0027] With this configuration, a method of manufacturing asemiconductor device can be provided with which even without any skilledoperator, the temperatures of all areas of a treatment target can besimply and promptly adjusted (soaking control) to a target value whilereducing resulting errors.

BRIEF DESCRIPTION OF THE DRAWINGS

[0028]FIG. 1 is a diagram showing an embodiment of a vertical diffusionfurnace to which a soaking control method for thermal treatmentapparatuses according to the present invention has been applied;

[0029]FIG. 2 is an enlarged view showing an interior of an outer wall ofthe vertical diffusion furnace, shown in FIG. 1;

[0030]FIG. 3 is a diagram showing an example of a profile thermocouple;

[0031]FIG. 4 is a diagram showing an example of a cascade thermocouple;

[0032]FIG. 5A is a side view showing a sheet-feed apparatus;

[0033]FIG. 5B is a top view showing heating zones of the sheet-feedapparatus;

[0034]FIG. 6 is a diagram showing a vertical diffusion furnace, which isa typical example of a known thermal treatment apparatus;

[0035]FIG. 7 is a diagram showing that thermocouple-mounted wafers arearranged so as to correspond to heating zones for wafers housed in thevertical diffusion furnace in FIG. 6;

[0036]FIG. 8 is a diagram showing that the thermocouple-mounted wafersare arranged so as to correspond to the heating zones for wafers housedin the vertical diffusion furnace in FIG. 6 and to zones between theheating zones;

[0037]FIG. 9 is a chart showing a relationship between temperaturesdetected by cascade thermocouples and the thermocouple-mounted wafers inthe thermal treatment apparatus set as in FIG. 7, wherein temperaturecontrol is carried out so that temperatures detected by the cascadethermocouples equal a target temperature for wafer treatment;

[0038]FIG. 10 is a chart showing a relationship between temperaturesdetected by the cascade thermocouples and the thermocouple-mountedwafers, wherein in order to improve the state shown in FIG. 9, thetemperature control is carried out so that the temperatures detected bythe cascade thermocouples equal a corrected target temperature;

[0039]FIG. 11 is a chart showing a relationship between temperaturesdetected by the cascade thermocouples and the thermocouple-mountedwafers, wherein in the state shown in FIG. 10, additionalthermocouple-mounted wafers are arranged at locations other than thosecorresponding to the heating zones for wafers; and

[0040]FIG. 12 is a chart showing a relationship between temperaturesdetected by the cascade thermocouples and the thermocouple-mountedwafers, wherein in order to improve the state shown in FIG. 11, thetemperature control is carried out so that the temperatures detected bythe cascade thermocouples equal a target value selected by a skilledoperator.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0041] Now, preferred embodiments of the present invention will bedescribed below with reference to the accompanying drawings.

[0042] The most fundamental form of the present invention is atemperature control method of controlling a heating apparatus having atleast one heating zone so that a temperature detected at a predeterminedlocation equals its target value, wherein the heating apparatus iscontrolled so that temperature is detected at a plurality ofpredetermined locations, the number of which is larger than that of theheating zones and that the target temperature falls between a maximumvalue and a minimum value of a plurality of detected temperatures. Inthe embodiment described below, a form will be described in which usingthe above form as a basic construction, temperature control is carriedout so that the average of differences between the plurality of detectedtemperatures and the target temperature is minimized. Embodiment 1.

[0043]FIG. 1 is a diagram showing an embodiment of a vertical diffusionfurnace to which a soaking control method for thermal treatmentapparatuses according to the present invention has been applied. FIG. 2is an enlarged view showing the interior of an outer wall of thevertical diffusion furnace, shown in FIG. 1. A vertical diffusionfurnace 10, shown in FIGS. 1 and 2, is composed of a soaking pipe 12 anda reaction pipe 13 both installed in an outer wall 11, a heater 14 forheating the interior of the furnace, heater thermocouples 15 a, 15 b, 15c, and 15 d for detecting the temperatures of the heater 14 at aplurality of different locations thereof, cascade thermocouples 16 a, 16b, 16 c, and 16 d for detecting the temperatures between the soakingpipe 12 and the reaction pipe 13, a boat 17 having a plurality of wafersand including thermocouple-mounted wafers 18 a, 18 a′, 18 b, 18 b′, 18b″, 18 c, 18 c′, and 18 d″ for detecting wafer temperatures(temperatures of a wafer and the area in which the wafer is arranged),and a temperature controller 19 for detecting an amount of operation Z(value of electric power) for the heater 14 on the basis of thetemperatures detected by the heater thermocouples 15 a, 15 b, 15 c, and15 d and the cascade thermocouples 16 a, 16 b, 16 c, and 16 d as well asa target temperature Y. In the above construction, the cascadethermocouples 16 a, 16 b, 16 c, and 16 d constitute first detectors ofthe present invention, and the thermocouple-mounted wafers 18 a, 18 a′,18 b, 18 b′, 18 b″, 18 c, 18 c′, and 18 d″ constitute second temperaturedetectors of the present invention.

[0044] In the above described example, the interior of the furnace isdivided into four heating zones, that is, U, CU, CL, and L zones fromthe top to the bottom of the figure in order to accurately control thetemperatures of wafer areas in the furnace. In each zone, high frequencypower can be applied to between heater terminals. The frequency powercan be varied for each zone. The temperature can be adjusted so as to beuniform among all the zones or to increase linearly among them. Theheater thermocouples 15 a, 15 b, 15 c, and 15 d and the cascadethermocouples 16 a, 16 b, 16 c, and 16 d are installed so as tocorrespond to the U, CU, CL, and L zones. Further, of the eight wafersmounted on the boat 17, four thermocouple-mounted wafers (18 a, 18 b, 18c, and 18 d) are arranged at locations representative of the U, CU, CL,and L zones, two thermocouple-mounted wafers are arranged between the Uand CU zones and between the CL and L zones, respectively, and twothermocouple-mounted wafers are arranged between CU and CL zones.

[0045] In a method of adjusting the temperatures of the wafer areas inthe vertical diffusion furnace 10 so as to be uniform, in preparationfor actual manufacturing of products (semiconductor devices), correctivevalues for a target temperature for temperature control executed duringthe manufacture are obtained. In order to make the temperatures of thewafer areas uniform, these corrective values for the target temperatureare used for the target value for the temperatures detected by thecascade thermocouples. That is, in the temperature control executedduring the actual manufacture, ordinary wafers for products are arrangedat the locations of the thermocouple-mounted wafers, so that thetemperatures of the wafers themselves cannot be detected. Thus, thetemperature controller 19 can execute temperature control such thattemperatures of the wafer areas are made uniform by applying thecorrective values to the target value for the temperatures detected bythe always arranged cascade thermocouples 16 a, 16 b, 16 c, and 16 d. Ofcourse, if the system is constructed to always measure the temperaturesof the wafer areas using a certain method, it should be appreciated thatthe control performance can be improved by controlling the temperaturesof the wafer areas instead of the cascade thermocouples.

[0046] Now, the principle of the method of adjusting the temperatures ofthe wafer areas of the vertical diffusion furnace 10 so as to be uniformwill be sequentially described. First, it is necessary to understand therelationship between the temperatures detected by the cascadethermocouples 16 a, 16 b, 16 c, and 16 d, used for control, and thetemperatures detected by the thermocouple-mounted wafers 18 a, 18 a′, 18b, 18 b′, 18 b″, 18 c, 18 c′and 18 d, arranged to detect thetemperatures of the wafer areas and which are the targets of the soakingcontrol. In the above described vertical diffusion furnace 10, thetemperatures detected by the eight thermocouple-mounted wafers areaffected by the heater 14, the temperature of which is detected by theheater thermocouples 15 a, 15 b, 15 c, and 15 d, corresponding to the U,CU, CL, and L zones, the four heating zones. The level of this effect isnumerically determined in a manner described below.

[0047] First, the system is controlled so that the temperatures detectedby the cascade thermocouples 16 a, 16 b, 16 c, and 16 d, correspondingto the U, CU, CL, and L zones, the four heating zones, equal the targetvalue for the wafer treatment. At this time, these temperatures need notbe exactly the same as the target value, but since the variation of thetemperature normally exhibits different characteristics depending ontemperature zones subjected to the control, the level of the effect mustbe numerically determined using temperature zones that prevent thetemperatures from significantly deviating from the target value. Aftertemperature has become stable in all the zones, several 0° C. (forexample, 10° C.) is added to the target temperature for the cascadethermocouple for one of the zones, for example, the U zone.Subsequently, after a sufficient time to stabilize the temperature haspassed, variations in the temperatures detected by the eightthermocouple-mounted wafers are recorded (a positive value is recordedif the temperature has increased, whereas a negative value is recordedif the temperature has decreased). On the basis of these results, thevariation added to the target temperature for the cascade thermocouplefor the U zone is defined as ΔT_(U), and the variations in thetemperatures detected by the eight thermocouple-mounted wafers at thistime are defined as ΔP_(U1) to ΔP_(U8) from the top to the bottom of thefurnace. Then, the following equations are obtained:

ΔP _(U1)=α_(U1) ×ΔT _(U)

ΔP _(U2)=α_(U2) ×ΔT _(U)

. . . =. . .

. . . =. . .

ΔP _(U8)=α_(U8) ×ΔT _(U)

[0048] where α_(U1) to α_(U8) denote coefficients indicating the levelof the effect of a variation in the target value for the cascadethermocouple for the U zone on the temperatures detected by the eightthermocouple-mounted wafers; these coefficients indicate that the largerthe numerical values are, the larger the effect is. At the same time,the variation added to the target temperature for the cascadethermocouple for the CU zone is defined as ΔT_(CU), and the variationsin the temperatures detected by the eight thermocouple-mounted wafers 18at this time are defined as ΔP_(CU1) to ΔP_(CU8) from the top to thebottom of the furnace. Then, the variations are expressed as follows:

ΔP _(CU1)=α_(CU1) ×ΔT _(CU)

ΔP _(CU2)=α_(CU2) ×ΔT _(CU)

. . . =. . .

. . . =. . .

ΔP _(CU8)=α_(CU8) ×ΔT _(CU)

[0049] For the CL zone, the variations are expressed as follows:

ΔP _(CL1)=α_(CL1) ×ΔT _(CL)

ΔP _(CL2)=α_(CL2) ×ΔT _(CL)

. . . =. . .

. . . =. . .

ΔP _(CL8)=α_(CL8) ×ΔT _(CL)

[0050] For the L zone, the variations are expressed as follows:

ΔP _(L1)=α_(L1) ×ΔT _(L)

ΔP _(L2)=α_(L2) ×ΔT _(L)

. . . =. . .

. . . =. . .

ΔP _(L8)=α_(L8) ×ΔT _(L)

[0051] On the basis of the above results, when the variations in thetemperatures detected by the eight thermocouple-mounted wafers aredefined as ΔP₁ to ΔP₈, the following equation is given:

ΔP ₁=(α_(U1) ×ΔT _(U))+(α_(CU1) ×ΔT _(CU))+(α_(CL1) ×ΔT _(CL))+(α_(L1)×ΔT _(L))

ΔP ₈=(α_(U8) ×ΔT _(U))+(α_(CU8) ×ΔT _(CU))+(α_(CL8) ×ΔT _(CL))+(α_(L8)×ΔT _(L))

[0052] Thus, the variations in the temperatures detected by the eightthermocouple-mounted wafers can be expressed by the variation in thetarget temperature for the cascade thermocouple for each zone multipliedby the corresponding coefficient. Accordingly, to bring the temperaturesdetected by the eight thermocouple-mounted wafers, close to the targetvalue, the above relational expressions must be used to determinedcorrective values for the target temperature for the cascadethermocouple for each zone. The above relational expressions can beexpressed as the following Equation (1): $\begin{matrix}{\begin{bmatrix}{\Delta \quad P_{1}} \\{\Delta \quad P_{2}} \\{\Delta \quad P_{3}} \\{\Delta \quad P_{4}} \\{\Delta \quad P_{5}} \\{\Delta \quad P_{6}} \\{\Delta \quad P_{7}} \\{\Delta \quad P_{8}}\end{bmatrix} = {\begin{bmatrix}\alpha_{U1} & \alpha_{CU1} & \alpha_{CL1} & \alpha_{L1} \\\alpha_{U2} & \alpha_{CU2} & \alpha_{CL2} & \alpha_{L2} \\\alpha_{U3} & \alpha_{CU3} & \alpha_{CL3} & \alpha_{L3} \\\alpha_{U4} & \alpha_{CU4} & \alpha_{CL4} & \alpha_{L4} \\\alpha_{U5} & \alpha_{CU5} & \alpha_{CL5} & \alpha_{L5} \\\alpha_{U6} & \alpha_{CU6} & \alpha_{CL6} & \alpha_{L6} \\\alpha_{U7} & \alpha_{CU7} & \alpha_{CL7} & \alpha_{L7} \\\alpha_{U8} & \alpha_{CU8} & \alpha_{CL8} & \alpha_{L8}\end{bmatrix} \times \begin{bmatrix}{\Delta \quad T_{U}} \\{\Delta \quad T_{CU}} \\{\Delta \quad T_{CL}} \\{\Delta \quad T_{L}}\end{bmatrix}}} & (1)\end{matrix}$

[0053] In the above Equation (1), the matrix in the first item of theright side is called an “interference matrix” (matrix of thecoefficients indicative of the level of the effect of the variation inthe target temperature for the cascade thermocouple on the temperaturesdetected by the thermocouple-mounted wafers) and defined as M, thecolumn vector (variation in the target temperature for the cascadethermocouple) in the second item of the right side is defined as ΔC, andthe column vector (variation in the temperature detected by thethermocouple-mounted wafer) in the left side is defined as ΔP. The aboveequation (1) can be expressed as follows:

ΔP=M×ΔC  (2)

[0054] In this manner, the relationship between the temperaturesdetected by the cascade thermocouples, used for control, and thetemperatures detected by the thermocouple-mounted wafers, the targets,can be numerically expressed. These numerical values, however, containerrors due to various factors. Accordingly, it should be appreciatedthat if accuracy must be improved or a sufficient adjustment time isavailable, the above described interference matrix may be repeatedlycreated several times to obtain the average value.

[0055] Now, before actual adjustment, the errors between thetemperatures detected by the thermocouple-mounted wafers and the targettemperature are obtained, which errors occur if the temperaturesdetected by the cascade thermocouples are controlled so as to equal thetarget temperature for the wafer treatment. Then, if the temperaturedetected by the thermocouple-mounted wafer is higher than the targetvalue, the error is expressed as a positive value. If this temperatureis lower than the target value, the error is expressed as a negativevalue. The errors between the temperatures detected by the eightthermocouple-mounted wafers and the target temperature are defined as E₁to E₈ from the top to the bottom heating zone, and can be expressed as amatrix vector P₀ such as the following Equation (3): $\begin{matrix}{P_{0} = \begin{bmatrix}E_{1} \\E_{2} \\E_{3} \\E_{4} \\E_{5} \\E_{6} \\E_{7} \\E_{8}\end{bmatrix}} & (3)\end{matrix}$

[0056] Next, adjustment is executed to actually bring the temperaturesdetected by the eight thermocouple-mounted wafers 18 a, 18 a′, 18 b, 18b′, 18 b″, 18 c, 18 c′, and 18 d, close to the target value. Before theadjustment, an evaluation criterion must be provided which is used toevaluate how the temperatures detected by the eight thermocouple-mountedwafers, the targets, have been brought close to the target value. Then,in order to increase the number of products having a quality of a fixedlevel or higher, the errors between the temperature detected by thethermocouple-mounted wafer and the target temperature must be minimizedfor all the thermocouple-mounted wafers. Thus, an evaluation expressionis obtained by squaring and then summing the temperatures detected bythe thermocouple-mounted wafers and the target temperature, and thesystem is controlled so as to minimize the evaluation expression. Ifadjustment is executed to minimize another evaluation criterion, forexample, the sum of the absolute values of the errors, the methoddescribed below will be used. The above described evaluation expressionis given below.

J=|P ₀ +ΔP| ²  (4)

[0057] The right side of this evaluation expression J represents the sumof the errors P₀ between the temperatures detected by the eightthermocouple-mounted wafers and the target temperature, which errors arepresent before the adjustment, and the adjustment-induced variations ΔPin the temperatures detected by the eight thermocouple-mounted wafers;it represents the errors between the temperatures detected by the eightthermocouple-mounted wafers and the target temperature, which errors arepresent after the adjustment. The above described Equation (2) isapplied to the evaluation expression J (Equation (4)) as follows.

J=|P ₀ +[M×ΔC]| ²  (5)

[0058] Then, matrix transposition is used.

J=[P ₀ +[M×ΔC]] ^(T) ×[P ₀ +[M×ΔC]]  (6)

[0059] where “^(T)” represents the matrix transposition. Furthermore,the above described Equations (1) and (3) are used to obtain thefollowing Equation (7). $\begin{matrix}\begin{matrix}{J = \quad {\left\lbrack {\begin{bmatrix}E_{1} \\E_{2} \\E_{3} \\E_{4} \\E_{5} \\E_{6} \\E_{7} \\E_{8}\end{bmatrix} + {\begin{bmatrix}\alpha_{U1} & \alpha_{CU1} & \alpha_{CL1} & \alpha_{L1} \\\alpha_{U2} & \alpha_{CU2} & \alpha_{CL2} & \alpha_{L2} \\\alpha_{U3} & \alpha_{CU3} & \alpha_{CL2} & \alpha_{L3} \\\alpha_{U4} & \alpha_{CU4} & \alpha_{CL2} & \alpha_{L4} \\\alpha_{U5} & \alpha_{CU5} & \alpha_{CL2} & \alpha_{L5} \\\alpha_{U6} & \alpha_{CU6} & \alpha_{CL2} & \alpha_{L6} \\\alpha_{U7} & \alpha_{CU7} & \alpha_{CL2} & \alpha_{L7} \\\alpha_{U8} & \alpha_{CU8} & \alpha_{CL2} & \alpha_{L8}\end{bmatrix} \times \begin{bmatrix}{\Delta \quad T_{U}} \\{\Delta \quad T_{CU}} \\{\Delta \quad T_{CL}} \\{\Delta \quad T_{L}}\end{bmatrix}}} \right\rbrack^{T} \times \left\lbrack {\begin{bmatrix}E_{1} \\E_{2} \\E_{3} \\E_{4} \\E_{5} \\E_{6} \\E_{7} \\E_{8}\end{bmatrix} + {\left\lbrack \quad \begin{matrix}\alpha_{U1} & \alpha_{CU1} & \alpha_{CL1} & \alpha_{L1} \\\alpha_{U2} & \alpha_{CU2} & \alpha_{CL2} & \alpha_{L2} \\\alpha_{U3} & \alpha_{CU3} & \alpha_{CL2} & \alpha_{L3} \\\alpha_{U4} & \alpha_{CU4} & \alpha_{CL2} & \alpha_{L4} \\\alpha_{U5} & \alpha_{CU5} & \alpha_{CL2} & \alpha_{L5} \\\alpha_{U6} & \alpha_{CU6} & \alpha_{CL2} & \alpha_{L6} \\\alpha_{U7} & \alpha_{CU7} & \alpha_{CL2} & \alpha_{L7} \\\alpha_{U8} & \alpha_{CU8} & \alpha_{CL2} & \alpha_{L8}\end{matrix}\quad \right\rbrack \times \begin{bmatrix}{\Delta \quad T_{U}} \\{\Delta \quad T_{CU}} \\{\Delta \quad T_{CL}} \\{\Delta \quad T_{L}}\end{bmatrix}}} \right\rbrack}} \\{= \quad {\begin{bmatrix}{E_{1} + {\alpha_{U1} \times \Delta \quad T_{U}} + {\alpha_{CU1} \times \Delta \quad T_{CU}} + {\alpha_{CL1} \times \Delta \quad T_{CL}} + {\alpha_{L1} \times \Delta \quad T_{L}}} \\{E_{2} + {\alpha_{U2} \times \Delta \quad T_{U}} + {\alpha_{CU2} \times \Delta \quad T_{CU}} + {\alpha_{CL2} \times \Delta \quad T_{CL}} + {\alpha_{L2} \times \Delta \quad T_{L}}} \\{E_{3} + {\alpha_{U3} \times \Delta \quad T_{U}} + {\alpha_{CU3} \times \Delta \quad T_{CU}} + {\alpha_{CL3} \times \Delta \quad T_{CL}} + {\alpha_{L3} \times \Delta \quad T_{L}}} \\{E_{4} + {\alpha_{U4} \times \Delta \quad T_{U}} + {\alpha_{CU4} \times \Delta \quad T_{CU}} + {\alpha_{CL4} \times \Delta \quad T_{CL}} + {\alpha_{L4} \times \Delta \quad T_{L}}} \\{E_{5} + {\alpha_{U5} \times \Delta \quad T_{U}} + {\alpha_{CU5} \times \Delta \quad T_{CU}} + {\alpha_{CL5} \times \Delta \quad T_{CL}} + {\alpha_{L5} \times \Delta \quad T_{L}}} \\{E_{6} + {\alpha_{U6} \times \Delta \quad T_{U}} + {\alpha_{CU6} \times \Delta \quad T_{CU}} + {\alpha_{CL6} \times \Delta \quad T_{CL}} + {\alpha_{L6} \times \Delta \quad T_{L}}} \\{E_{7} + {\alpha_{U7} \times \Delta \quad T_{U}} + {\alpha_{CU7} \times \Delta \quad T_{CU}} + {\alpha_{CL7} \times \Delta \quad T_{CL}} + {\alpha_{L7} \times \Delta \quad T_{L}}} \\{E_{8} + {\alpha_{U8} \times \Delta \quad T_{U}} + {\alpha_{CU8} \times \Delta \quad T_{CU}} + {\alpha_{CL8} \times \Delta \quad T_{CL}} + {\alpha_{L8} \times \Delta \quad T_{L}}}\end{bmatrix}^{T} \times}} \\{\quad \begin{bmatrix}{E_{1} + {\alpha_{U1} \times \Delta \quad T_{U}} + {\alpha_{CU1} \times \Delta \quad T_{CU}} + {\alpha_{CL1} \times \Delta \quad T_{CL}} + {\alpha_{L1} \times \Delta \quad T_{L}}} \\{E_{2} + {\alpha_{U2} \times \Delta \quad T_{U}} + {\alpha_{CU2} \times \Delta \quad T_{CU}} + {\alpha_{CL2} \times \Delta \quad T_{CL}} + {\alpha_{L2} \times \Delta \quad T_{L}}} \\{E_{3} + {\alpha_{U3} \times \Delta \quad T_{U}} + {\alpha_{CU3} \times \Delta \quad T_{CU}} + {\alpha_{CL3} \times \Delta \quad T_{CL}} + {\alpha_{L3} \times \Delta \quad T_{L}}} \\{E_{4} + {\alpha_{U4} \times \Delta \quad T_{U}} + {\alpha_{CU4} \times \Delta \quad T_{CU}} + {\alpha_{CL4} \times \Delta \quad T_{CL}} + {\alpha_{L4} \times \Delta \quad T_{L}}} \\{E_{5} + {\alpha_{U5} \times \Delta \quad T_{U}} + {\alpha_{CU5} \times \Delta \quad T_{CU}} + {\alpha_{CL5} \times \Delta \quad T_{CL}} + {\alpha_{L5} \times \Delta \quad T_{L}}} \\{E_{6} + {\alpha_{U6} \times \Delta \quad T_{U}} + {\alpha_{CU6} \times \Delta \quad T_{CU}} + {\alpha_{CL6} \times \Delta \quad T_{CL}} + {\alpha_{L6} \times \Delta \quad T_{L}}} \\{E_{7} + {\alpha_{U7} \times \Delta \quad T_{U}} + {\alpha_{CU7} \times \Delta \quad T_{CU}} + {\alpha_{CL7} \times \Delta \quad T_{CL}} + {\alpha_{L7} \times \Delta \quad T_{L}}} \\{E_{8} + {\alpha_{U8} \times \Delta \quad T_{U}} + {\alpha_{CU8} \times \Delta \quad T_{CU}} + {\alpha_{CL8} \times \Delta \quad T_{CL}} + {\alpha_{L8} \times \Delta \quad T_{L}}}\end{bmatrix}} \\{= \quad \begin{matrix}{\left( {E_{1} + {\alpha_{U1} \times \Delta \quad T_{U}} + {\alpha_{CU1} \times \Delta \quad T_{CU}} + {\alpha_{CL1} \times \Delta \quad T_{CL}} + {\alpha_{L1} \times \Delta \quad T_{L}}} \right)^{2} +} \\{\left( {E_{2} + {\alpha_{U2} \times \Delta \quad T_{U}} + {\alpha_{CU2} \times \Delta \quad T_{CU}} + {\alpha_{CL2} \times \Delta \quad T_{CL}} + {\alpha_{L2} \times \Delta \quad T_{L}}} \right)^{2} +} \\{\left( {E_{3} + {\alpha_{U3} \times \Delta \quad T_{U}} + {\alpha_{CU3} \times \Delta \quad T_{CU}} + {\alpha_{CL3} \times \Delta \quad T_{CL}} + {\alpha_{L3} \times \Delta \quad T_{L}}} \right)^{2} +} \\{\left( {E_{4} + {\alpha_{U4} \times \Delta \quad T_{U}} + {\alpha_{CU4} \times \Delta \quad T_{CU}} + {\alpha_{CL4} \times \Delta \quad T_{CL}} + {\alpha_{L4} \times \Delta \quad T_{L}}} \right)^{2} +} \\{\left( {E_{5} + {\alpha_{U5} \times \Delta \quad T_{U}} + {\alpha_{CU5} \times \Delta \quad T_{CU}} + {\alpha_{CL5} \times \Delta \quad T_{CL}} + {\alpha_{L5} \times \Delta \quad T_{L}}} \right)^{2} +} \\{\left( {E_{6} + {\alpha_{U6} \times \Delta \quad T_{U}} + {\alpha_{CU6} \times \Delta \quad T_{CU}} + {\alpha_{CL6} \times \Delta \quad T_{CL}} + {\alpha_{L6} \times \Delta \quad T_{L}}} \right)^{2} +} \\{\left( {E_{7} + {\alpha_{U7} \times \Delta \quad T_{U}} + {\alpha_{CU7} \times \Delta \quad T_{CU}} + {\alpha_{CL7} \times \Delta \quad T_{CL}} + {\alpha_{L7} \times \Delta \quad T_{L}}} \right)^{2} +} \\\left( {E_{8} + {\alpha_{U8} \times \Delta \quad T_{U}} + {\alpha_{CU8} \times \Delta \quad T_{CU}} + {\alpha_{CL8} \times \Delta \quad T_{CL}} + {\alpha_{L8} \times \Delta \quad T_{L}}} \right)^{2}\end{matrix}}\end{matrix} & (7)\end{matrix}$

[0060] Next, to determine the elements ΔT_(U), ΔT_(CU), ΔT_(CL), andΔT_(L) of ΔC in order to minimize the evaluation expression J, thisexpression is partially differentiated for the elements ΔT_(U), ΔT_(CU),ΔT_(CL), and ΔT_(L) of ΔC. First, the partial differentiation of theevaluation expression J for ΔTU results in the following Equation (8).$\begin{matrix}\begin{matrix}{\frac{\partial J}{{\partial\Delta}\quad T_{U}} = \begin{matrix}{\left( {\left( {2 \times \alpha_{U1}} \right) \times \left( {E_{1} + \left( {\alpha_{U1} + {\Delta \quad T_{U}}} \right) + \left( {\alpha_{CU1} \times \Delta \quad T_{CU}} \right) + \left( {\alpha_{CL1} \times \Delta \quad T_{CL}} \right) + \left( {\alpha_{L1} \times \Delta \quad T_{L}} \right)} \right)} \right) +} \\{\left( {\left( {2 \times \alpha_{U2}} \right) \times \left( {E_{2} + \left( {\alpha_{U2} + {\Delta \quad T_{U}}} \right) + \left( {\alpha_{CU2} \times \Delta \quad T_{CU}} \right) + \left( {\alpha_{CL2} \times \Delta \quad T_{CL}} \right) + \left( {\alpha_{L2} \times \Delta \quad T_{L}} \right)} \right)} \right) +} \\{\left( {\left( {2 \times \alpha_{U3}} \right) \times \left( {E_{3} + \left( {\alpha_{U3} + {\Delta \quad T_{U}}} \right) + \left( {\alpha_{CU3} \times \Delta \quad T_{CU}} \right) + \left( {\alpha_{CL3} \times \Delta \quad T_{CL}} \right) + \left( {\alpha_{L3} \times \Delta \quad T_{L}} \right)} \right)} \right) +} \\{\left( {\left( {2 \times \alpha_{U4}} \right) \times \left( {E_{4} + \left( {\alpha_{U4} + {\Delta \quad T_{U}}} \right) + \left( {\alpha_{CU4} \times \Delta \quad T_{CU}} \right) + \left( {\alpha_{CL4} \times \Delta \quad T_{CL}} \right) + \left( {\alpha_{L4} \times \Delta \quad T_{L}} \right)} \right)} \right) +} \\{\left( {\left( {2 \times \alpha_{U5}} \right) \times \left( {E_{5} + \left( {\alpha_{U5} + {\Delta \quad T_{U}}} \right) + \left( {\alpha_{CU5} \times \Delta \quad T_{CU}} \right) + \left( {\alpha_{CL5} \times \Delta \quad T_{CL}} \right) + \left( {\alpha_{L5} \times \Delta \quad T_{L}} \right)} \right)} \right) +} \\{\left( {\left( {2 \times \alpha_{U6}} \right) \times \left( {E_{6} + \left( {\alpha_{U6} + {\Delta \quad T_{U}}} \right) + \left( {\alpha_{CU6} \times \Delta \quad T_{CU}} \right) + \left( {\alpha_{CL6} \times \Delta \quad T_{CL}} \right) + \left( {\alpha_{L6} \times \Delta \quad T_{L}} \right)} \right)} \right) +} \\{\left( {\left( {2 \times \alpha_{U7}} \right) \times \left( {E_{7} + \left( {\alpha_{U7} + {\Delta \quad T_{U}}} \right) + \left( {\alpha_{CU7} \times \Delta \quad T_{CU}} \right) + \left( {\alpha_{CL7} \times \Delta \quad T_{CL}} \right) + \left( {\alpha_{L7} \times \Delta \quad T_{L}} \right)} \right)} \right) +} \\\left( {\left( {2 \times \alpha_{U8}} \right) \times \left( {E_{8} + \left( {\alpha_{U8} + {\Delta \quad T_{U}}} \right) + \left( {\alpha_{CU8} \times \Delta \quad T_{CU}} \right) + \left( {\alpha_{CL8} \times \Delta \quad T_{CL}} \right) + \left( {\alpha_{L8} \times \Delta \quad T_{L}} \right)} \right)} \right)\end{matrix}} \\{= \begin{matrix}{2 \times \left\{ {{\left( {\alpha_{U1}^{2} + \alpha_{U2}^{2} + \alpha_{U3}^{2} + \alpha_{U4}^{2} + \alpha_{U5}^{2} + \alpha_{U6}^{2} + \alpha_{U7}^{2} + \alpha_{U8}^{2}} \right) \times \Delta \quad T_{U}} +} \right.} \\\left( {\left( {\alpha_{U1} \times \alpha_{CU1}} \right) + \left( {\alpha_{U2} \times \alpha_{CU2}} \right) + \left( {\alpha_{U3} \times \alpha_{CU3}} \right) + \left( {\alpha_{U4} \times \alpha_{CU4}} \right) +} \right. \\{{\left. {\left( {\alpha_{U5} \times \alpha_{CU5}} \right) + \left( {\alpha_{U6} \times \alpha_{CU6}} \right) + \left( {\alpha_{U7} \times \alpha_{CU7}} \right) + \left( {\alpha_{U8} \times \alpha_{CU8}} \right)} \right) \times \Delta \quad T_{CU}} +} \\{{\left. {\left( {\alpha_{U5} \times \alpha_{CL5}} \right) + \left( {\alpha_{U6} \times \alpha_{CL6}} \right) + \left( {\alpha_{U7} \times \alpha_{CL7}} \right) + \left( {\alpha_{U8} \times \alpha_{CL8}} \right)} \right) \times \Delta \quad T_{CL}} +} \\\left( {\left( {\alpha_{U1} \times \alpha_{L1}} \right) + \left( {\alpha_{U2} \times \alpha_{L2}} \right) + \left( {\alpha_{U3} \times \alpha_{L3}} \right) + \left( {\alpha_{U4} \times \alpha_{L4}} \right) +} \right. \\{{\left. {\left( {\alpha_{U5} \times \alpha_{L5}} \right) + \left( {\alpha_{U6} \times \alpha_{L6}} \right) + \left( {\alpha_{U7} \times \alpha_{L7}} \right) + \left( {\alpha_{U8} \times \alpha_{L8}} \right)} \right) \times \Delta \quad T_{L}} +} \\\left( {\left( {\alpha_{U1} \times E_{1}} \right) + \left( {\alpha_{U2} \times E_{2}} \right) + \left( {\alpha_{U3} \times E_{3}} \right) + \left( {\alpha_{U4} \times E_{4}} \right) +} \right. \\\left. \left. {\left( {\alpha_{U5} \times E_{5}} \right) + \left( {\alpha_{U6} \times E_{6}} \right) + \left( {\alpha_{U7} \times E_{7}} \right) + \left( {\alpha_{U8} \times E_{8}} \right)} \right) \right\}\end{matrix}}\end{matrix} & (8)\end{matrix}$

[0061] The evaluation expression J is then partially differentiated forthe elements Δ_(CU), ΔT_(CL), and ΔT_(L) as described above to obtainthe following Equations (9), (10), and (11). $\begin{matrix}{\frac{\partial J}{{\partial\Delta}\quad T_{CU}} = {2 \times \left\{ {{\left( {\left( {\alpha_{U1} \times \alpha_{CU1}} \right) + \left( {\alpha_{U2} \times \alpha_{CU2}} \right) + \left( {\alpha_{U3} \times \alpha_{CU3}} \right) + \left( {\alpha_{U4} \times \alpha_{CU4}} \right) + \left( {\alpha_{U5} \times \alpha_{CU5}} \right) + \left( {\alpha_{U6} \times \alpha_{CU6}} \right) + \left( {\alpha_{U7} \times \alpha_{CU7}} \right) + \left( {\alpha_{U8} \times \alpha_{CU8}} \right)} \right) \times \Delta \quad T_{U}} + {\left( {\alpha_{CU1}^{2} + \alpha_{CU2}^{2} + \alpha_{CU3}^{2} + \alpha_{CU4}^{2} + \alpha_{CU5}^{2} + \alpha_{CU6}^{2} + \alpha_{CU7}^{2} + \alpha_{CU8}^{2}} \right) \times \Delta \quad T_{CU}} + {\left( {\left( {\alpha_{CU1} \times \alpha_{CL1}} \right) + \left( {\alpha_{CU2} \times \alpha_{CL2}} \right) + \left( {\alpha_{CU3} \times \alpha_{CL3}} \right) + \left( {\alpha_{CU4} \times \alpha_{CL4}} \right) + \left( {\alpha_{CU5} \times \alpha_{CL5}} \right) + \left( {\alpha_{CU6} \times \alpha_{CL6}} \right) + \left( {\alpha_{CU7} \times \alpha_{CL7}} \right) + \left( {\alpha_{CU8} \times \alpha_{CL8}} \right)} \right) \times \Delta \quad T_{CL}} + {\left( {\left( {\alpha_{CU1} \times \alpha_{L1}} \right) + \left( {\alpha_{CU2} \times \alpha_{L2}} \right) + \left( {\alpha_{CU3} \times \alpha_{L3}} \right) + \left( {\alpha_{CU4} \times \alpha_{L4}} \right) + \left( {\alpha_{CU5} \times \alpha_{L5}} \right) + \left( {\alpha_{CU6} \times \alpha_{L6}} \right) + \left( {\alpha_{CU7} + \alpha_{L7}} \right) + \left( {\alpha_{CU8} \times \alpha_{L8}} \right)} \right) \times \Delta \quad T_{L}} + \left( {\left( {\alpha_{CU1} \times E_{1}} \right) + \left( {\alpha_{CU2} \times E_{2}} \right) + \left( {\alpha_{CU3} \times E_{3}} \right) + \left( {\alpha_{CU4} \times E_{4}} \right) + \left( {\alpha_{CU5} \times E_{5}} \right) + \left( {\alpha_{CU6} \times E_{6}} \right) + \left( {\alpha_{CU7} \times E_{7}} \right) + \left( {\alpha_{CU8} \times E_{8}} \right)} \right)} \right\}}} & (9) \\{\frac{\partial J}{{\partial\Delta}\quad T_{CL}} = {2 \times \left\{ {{\left( {\left( {\alpha_{U1} \times \alpha_{CL1}} \right) + \left( {\alpha_{U2} \times \alpha_{CL2}} \right) + \left( {\alpha_{U3} \times \alpha_{CL3}} \right) + \left( {\alpha_{U4} \times \alpha_{CL4}} \right) + \left( {\alpha_{U5} \times \alpha_{CL5}} \right) + \left( {\alpha_{U6} \times \alpha_{CL6}} \right) + \left( {\alpha_{U7} + \alpha_{CL7}} \right) + \left( {\alpha_{U8} \times \alpha_{CL8}} \right)} \right) \times \Delta \quad T_{U}} + {\left( {\left( {\alpha_{CU1} + \alpha_{CL1}} \right) + \left( {\alpha_{CU2} + \alpha_{CL2}} \right) + \left( {\alpha_{CU3} \times \alpha_{CL3}} \right) + \left( {\alpha_{CU4} \times \alpha_{CL4}} \right) + \left( {\alpha_{CU5} \times \alpha_{CL5}} \right) + \left( {\alpha_{CU6} \times \alpha_{CL6}} \right) + \left( {\alpha_{CU7} \times \alpha_{CL7}} \right) + \left( {\alpha_{CU8} \times \alpha_{CL8}} \right)} \right) \times \Delta \quad T_{CU}} + {\left( {\alpha_{CL1}^{2} + \alpha_{CL2}^{2} + \alpha_{CL3}^{2} + \alpha_{CL4}^{2} + \alpha_{CL5}^{2} + \alpha_{CL6}^{2} + \alpha_{CL7}^{2} + \alpha_{CL8}^{2}} \right) \times \Delta \quad T_{CL}} + {\left( {\left( {\alpha_{CL1} \times \alpha_{L1}} \right) + \left( {\alpha_{CL2} \times \alpha_{L2}} \right) + \left( {\alpha_{CL3} \times \alpha_{L3}} \right) + \left( {\alpha_{CL4} \times \alpha_{L4}} \right) + \left( {\alpha_{CL5} \times \alpha_{L5}} \right) + \left( {\alpha_{CL6} \times \alpha_{L6}} \right) + \left( {\alpha_{CL7} \times \alpha_{L7}} \right) + \left( {\alpha_{CL8} \times \alpha_{L8}} \right)} \right) \times \Delta \quad T_{L}} + \left( {\left( {\alpha_{CL1} \times E_{1}} \right) + \left( {\alpha_{CL2} \times E_{2}} \right) + \left( {\alpha_{CL3} \times E_{3}} \right) + \left( {\alpha_{CL4} \times E_{4}} \right) + \left( {\alpha_{CL5} \times E_{5}} \right) + \left( {\alpha_{CL6} \times E_{6}} \right) + \left( {\alpha_{CL7} \times E_{7}} \right) + \left( {\alpha_{CL8} \times E_{8}} \right)} \right)} \right\}}} & (10) \\{\frac{\partial J}{{\partial\Delta}\quad T_{L}} = {2 \times \left\{ {{\left( {\left( {\alpha_{U1} \times \alpha_{L1}} \right) + \left( {\alpha_{U2} \times \alpha_{L2}} \right) + \left( {\alpha_{U3} \times \alpha_{L3}} \right) + \left( {\alpha_{U4} \times \alpha_{L4}} \right) + \left( {\alpha_{U5} \times \alpha_{L5}} \right) + \left( {\alpha_{U6} \times \alpha_{L6}} \right) + \left( {\alpha_{U7} + \alpha_{L7}} \right) + \left( {\alpha_{U8} \times \alpha_{L8}} \right)} \right) \times \Delta \quad T_{U}} + {\left( {\left( {\alpha_{CU1} + \alpha_{L1}} \right) + \left( {\alpha_{CU2} + \alpha_{L2}} \right) + \left( {\alpha_{CU3} \times \alpha_{L3}} \right) + \left( {\alpha_{CU4} \times \alpha_{L4}} \right) + \left( {\alpha_{CU5} \times \alpha_{L5}} \right) + \left( {\alpha_{CU6} \times \alpha_{L6}} \right) + \left( {\alpha_{CU7} \times \alpha_{L7}} \right) + \left( {\alpha_{CU8} \times \alpha_{L8}} \right)} \right) \times \Delta \quad T_{CU}} + {\left( {\left( {\alpha_{CL1} \times \alpha_{L1}} \right) + \left( {\alpha_{CL2} \times \alpha_{L2}} \right) + \left( {\alpha_{CL3} \times \alpha_{L3}} \right) + \left( {\alpha_{CL4} \times \alpha_{L4}} \right) + \left( {\alpha_{CL5} \times \alpha_{L5}} \right) + \left( {\alpha_{CL6} \times \alpha_{L6}} \right) + \left( {\alpha_{CL7} \times \alpha_{L7}} \right) + \left( {\alpha_{CL8} \times \alpha_{L8}} \right)} \right) \times \Delta \quad T_{CL}} + {\left( {\alpha_{L1}^{2} + \alpha_{L2}^{2} + \alpha_{L3}^{2} + \alpha_{L4}^{2} + \alpha_{L5}^{2} + \alpha_{L6}^{2} + \alpha_{L7}^{2} + \alpha_{L8}^{2}} \right) \times \Delta \quad T_{L}} + \left( {\left( {\alpha_{L1} \times E_{1}} \right) + \left( {\alpha_{L2} \times E_{2}} \right) + \left( {\alpha_{L3} \times E_{3}} \right) + \left( {\alpha_{L4} \times E_{4}} \right) + \left( {\alpha_{L5} \times E_{5}} \right) + \left( {\alpha_{L6} \times E_{6}} \right) + \left( {\alpha_{L7} \times E_{7}} \right) + \left( {\alpha_{L8} \times E_{8}} \right)} \right)} \right\}}} & (11)\end{matrix}$

[0062] Thus, on the basis of the results of the partial differentiationof the elements ΔT_(U), ΔT_(CU), ΔT_(CL), and ΔT_(L) of ΔT, thefollowing equations are given. ∂J/∂Δ  T_(U) = 0 ∂J/∂Δ  T_(CU) = 0∂J/∂Δ  T_(CL) = 0 ∂J/∂Δ  T_(L) = 0

[0063] These are four-element linear equation containing the elementsΔT_(U), ΔT_(CU), ΔT_(CL), and ΔT_(L) of ΔC as variables, and the resultsΔT_(U), ΔT_(CU), ΔT_(CL), and ΔT_(L) obtained by solving thesesimultaneous equations minimize the evaluation expression J. That is,these are corrective values for the target temperature for the cascadethermocouples which values minimize the sum of the square of thedifferences between the temperature detected by the thermocouple-mountedwafers and the target temperature.

[0064] Now, a method of solving the above described simultaneousfour-element linear equations will be shown. First, the equation∂J/∂ΔT_(U)=0 can be expressed as the following Equation (12).$\begin{matrix}{{\begin{bmatrix}{\alpha_{U1}^{2} + \ldots + \alpha_{U8}^{2}} \\{\left( {\alpha_{U1} \times \alpha_{CU1}} \right) + \ldots + \left( {\alpha_{U8} \times \alpha_{CU8}} \right)} \\{\left( {\alpha_{U1} \times \alpha_{CL1}} \right) + \ldots + \left( {\alpha_{U8} \times \alpha_{CL8}} \right)} \\{\left( {\alpha_{U1} \times \alpha_{L1}} \right) + \ldots + \left( {\alpha_{U8} \times \alpha_{L8}} \right)}\end{bmatrix}^{T} \times \begin{bmatrix}{\Delta \quad T_{U}} \\{\Delta \quad T_{CU}} \\{\Delta \quad T_{CL}} \\{\Delta \quad T_{L}}\end{bmatrix}} = {\left( {- 1} \right) \times \left( {\left( {\alpha_{U1} \times E_{1}} \right) + \ldots + \left( {\alpha_{U8} \times E_{8}} \right)} \right)}} & (12)\end{matrix}$

[0065] Likewise, the expressions ∂J/∂ΔT_(CU)=0, ∂J/∂ΔT_(CL)=0, and∂J/∂ΔT_(L)=0 can be expressed as the following Equations (13), (14), and(15). $\begin{matrix}{{\begin{bmatrix}{\left( {\alpha_{U1} \times \alpha_{CU1}} \right) + \ldots + \left( {\alpha_{U8} \times \alpha_{CU8}} \right)} \\{\alpha_{CU1}^{2} + \ldots + \alpha_{CU8}^{2}} \\{\left( {\alpha_{CU1} \times \alpha_{CL1}} \right) + \ldots + \left( {\alpha_{CU8} \times \alpha_{CL8}} \right)} \\{\left( {\alpha_{CU1} \times \alpha_{L1}} \right) + \ldots + \left( {\alpha_{CU8} \times \alpha_{L8}} \right)}\end{bmatrix}^{T} \times \begin{bmatrix}{\Delta \quad T_{U}} \\{\Delta \quad T_{CU}} \\{\Delta \quad T_{CL}} \\{\Delta \quad T_{L}}\end{bmatrix}} = {\left( {- 1} \right) \times \left( {\left( {\alpha_{CU1} \times E_{1}} \right) + \ldots + \left( {\alpha_{CU8} \times E_{8}} \right)} \right)}} & (13) \\{{\begin{bmatrix}{\left( {\alpha_{U1} \times \alpha_{CL1}} \right) + \ldots + \left( {\alpha_{U8} \times \alpha_{CL8}} \right)} \\{\left( {\alpha_{CU1} \times \alpha_{CL1}} \right) + \ldots + \left( {\alpha_{CU8} \times \alpha_{CL8}} \right)} \\{\alpha_{CL1}^{2} + \ldots + \alpha_{CL8}^{2}} \\{\left( {\alpha_{CL1} \times \alpha_{L1}} \right) + \ldots + \left( {\alpha_{CL8} \times \alpha_{L8}} \right)}\end{bmatrix}^{T} \times \begin{bmatrix}{\Delta \quad T_{U}} \\{\Delta \quad T_{CU}} \\{\Delta \quad T_{CL}} \\{\Delta \quad T_{L}}\end{bmatrix}} = {\left( {- 1} \right) \times \left( {\left( {\alpha_{CL1} \times E_{1}} \right) + \ldots + \left( {\alpha_{CL8} \times E_{8}} \right)} \right)}} & (14) \\{{\begin{bmatrix}{\left( {\alpha_{U1} \times \alpha_{L1}} \right) + \ldots + \left( {\alpha_{U8} \times \alpha_{L8}} \right)} \\{\left( {\alpha_{CU1} \times \alpha_{L1}} \right) + \ldots + \left( {\alpha_{CU8} \times \alpha_{L8}} \right)} \\{\left( {\alpha_{CL1} \times \alpha_{L1}} \right) + \ldots + \left( {\alpha_{CL8} \times \alpha_{L8}} \right)} \\{\alpha_{L1}^{2} + \ldots + \alpha_{L8}^{2}}\end{bmatrix}^{T} \times \begin{bmatrix}{\Delta \quad T_{U}} \\{\Delta \quad T_{CU}} \\{\Delta \quad T_{CL}} \\{\Delta \quad T_{L}}\end{bmatrix}} = {\left( {- 1} \right) \times \left( {\left( {\alpha_{L1} \times E_{1}} \right) + \ldots + \left( {\alpha_{L8} \times E_{8}} \right)} \right)}} & (15)\end{matrix}$

[0066] The above four equations can be expressed as the followingEquation (16) using matrices. $\begin{matrix}{{\begin{bmatrix}{\alpha_{U1}^{2} + \ldots + \alpha_{U8}^{2}} & {\left( {\alpha_{U1} \times \alpha_{CU1}} \right) + \ldots + \left( {\alpha_{U8} \times \alpha_{CU8}} \right)} & {\left( {\alpha_{U1} \times \alpha_{CL1}} \right) + \ldots + \left( {\alpha_{U8} \times \alpha_{CL8}} \right)} & {\left( {\alpha_{U1} \times \alpha_{L1}} \right) + \ldots + \left( {\alpha_{U8} \times \alpha_{L8}} \right)} \\{\left( {\alpha_{U1} \times \alpha_{CU1}} \right) + \ldots + \left( {\alpha_{U8} \times \alpha_{CU8}} \right)} & {\alpha_{CU1}^{2} + \ldots + \alpha_{CU8}^{2}} & {\left( {\alpha_{CU1} \times \alpha_{CL1}} \right) + \ldots + \left( {\alpha_{CU8} \times \alpha_{CL8}} \right)} & {\left( {\alpha_{CU1} \times \alpha_{L1}} \right) + \ldots + \left( {\alpha_{CU8} \times \alpha_{L8}} \right)} \\{\left( {\alpha_{U1} \times \alpha_{CL1}} \right) + \ldots + \left( {\alpha_{U8} \times \alpha_{CL8}} \right)} & {\left( {\alpha_{CU1} \times \alpha_{CL1}} \right) + \ldots + \left( {\alpha_{CU8} \times \alpha_{CL8}} \right)} & {\alpha_{CL1}^{2} + \ldots + \alpha_{CL8}^{2}} & {\left( {\alpha_{CL1} \times \alpha_{L1}} \right) + \ldots + \left( {\alpha_{CL8} \times \alpha_{L8}} \right)} \\{\left( {\alpha_{U1} \times \alpha_{L1}} \right) + \ldots + \left( {\alpha_{U8} \times \alpha_{L8}} \right)} & {\left( {\alpha_{CU1} \times \alpha_{L1}} \right) + \ldots + \left( {\alpha_{CU8} \times \alpha_{L8}} \right)} & {\left( {\alpha_{CL1} \times \alpha_{L1}} \right) + \ldots + \left( {\alpha_{CL8} \times \alpha_{L8}} \right)} & {\alpha_{L1}^{2} + {\ldots\alpha}_{L8}^{2}}\end{bmatrix} \times \quad \quad \left\lbrack \quad \begin{matrix}{\Delta \quad T} \\{\Delta \quad T_{CU}} \\{\Delta \quad T_{CL}} \\{\Delta \quad T_{L}}\end{matrix}\quad \right\rbrack} = \begin{bmatrix}{\left( {- 1} \right) \times \left( {\left( {\alpha_{U1} \times E_{1}} \right) + \ldots + \left( {\alpha_{U8} \times E_{8}} \right)} \right)} \\{\left( {- 1} \right) \times \left( {\left( {\alpha_{CU1} \times E_{1}} \right) + \ldots + \left( {\alpha_{CU8} \times E_{8}} \right)} \right)} \\{\left( {- 1} \right) \times \left( {\left( {\alpha_{CL1} \times E_{1}} \right) + \ldots + \left( {\alpha_{CL8} \times E_{8}} \right)} \right)} \\{\left( {- 1} \right) \times \left( {\left( {\alpha_{L1} \times E_{1}} \right) + \ldots + \left( {\alpha_{L8} \times E_{8}} \right)} \right)}\end{bmatrix}} & (16)\end{matrix}$

[0067] In the above Equation (16), the matrix in the first item of theleft side can be expressed as the following Equation (17) using theinterference matrix M in Equation (2). $\begin{matrix}{\begin{bmatrix}{\alpha_{U1}^{2} + \ldots + \alpha_{U8}^{2}} & {\left( {\alpha_{U1} \times \alpha_{CU1}} \right) + \ldots + \left( {\alpha_{U8} \times \alpha_{CU8}} \right)} & {\left( {\alpha_{U1} \times \alpha_{CL1}} \right) + \ldots + \left( {\alpha_{U8} \times \alpha_{CL8}} \right)} & {\left( {\alpha_{U1} \times \alpha_{L1}} \right) + \ldots + \left( {\alpha_{U8} \times \alpha_{L8}} \right)} \\{\left( {\alpha_{U1} \times \alpha_{CU1}} \right) + \ldots + \left( {\alpha_{U8} \times \alpha_{CU8}} \right)} & {\alpha_{CU1}^{2} + \ldots + \alpha_{CU8}^{2}} & {\left( {\alpha_{CU1} \times \alpha_{CL1}} \right) + \ldots + \left( {\alpha_{CU8} \times \alpha_{CL8}} \right)} & {\left( {\alpha_{CU1} \times \alpha_{L1}} \right) + \ldots + \left( {\alpha_{CU8} \times \alpha_{L8}} \right)} \\{\left( {\alpha_{U1} \times \alpha_{CL1}} \right) + \ldots + \left( {\alpha_{U8} \times \alpha_{CL8}} \right)} & {\left( {\alpha_{CU1} \times \alpha_{CL1}} \right) + \ldots + \left( {\alpha_{CU8} \times \alpha_{CL8}} \right)} & {\alpha_{CL1}^{2} + \ldots + \alpha_{CL8}^{2}} & {\left( {\alpha_{CL1} \times \alpha_{L1}} \right) + \ldots + \left( {\alpha_{CL8} \times \alpha_{L8}} \right)} \\{\left( {\alpha_{U1} \times \alpha_{L1}} \right) + \ldots + \left( {\alpha_{U8} \times \alpha_{L8}} \right)} & {\left( {\alpha_{CU1} \times \alpha_{L1}} \right) + \ldots + \left( {\alpha_{CU8} \times \alpha_{L8}} \right)} & {\left( {\alpha_{CL1} \times \alpha_{L1}} \right) + \ldots + \left( {\alpha_{CL8} \times \alpha_{L8}} \right)} & {\alpha_{L1}^{2} + {\ldots\alpha}_{L8}^{2}}\end{bmatrix} = {{\begin{bmatrix}\alpha_{U1} & \alpha_{CU1} & \alpha_{CL1} & \alpha_{L1} \\\alpha_{U2} & \alpha_{CU2} & \alpha_{CL2} & \alpha_{L2} \\\alpha_{U3} & \alpha_{CU3} & \alpha_{CL3} & \alpha_{L3} \\\alpha_{U4} & \alpha_{CU4} & \alpha_{CL4} & \alpha_{L4} \\\alpha_{U5} & \alpha_{CU5} & \alpha_{CL5} & \alpha_{L5} \\\alpha_{U6} & \alpha_{CU6} & \alpha_{CL6} & \alpha_{L6} \\\alpha_{U7} & \alpha_{CU7} & \alpha_{CL7} & \alpha_{L7} \\\alpha_{U8} & \alpha_{CU8} & \alpha_{CL8} & \alpha_{L8}\end{bmatrix}^{T} \times \begin{bmatrix}\alpha_{U1} & \alpha_{CU1} & \alpha_{CL1} & \alpha_{L1} \\\alpha_{U2} & \alpha_{CU2} & \alpha_{CL2} & \alpha_{L2} \\\alpha_{U3} & \alpha_{CU3} & \alpha_{CL3} & \alpha_{L3} \\\alpha_{U4} & \alpha_{CU4} & \alpha_{CL4} & \alpha_{L4} \\\alpha_{U5} & \alpha_{CU5} & \alpha_{CL5} & \alpha_{L5} \\\alpha_{U6} & \alpha_{CU6} & \alpha_{CL6} & \alpha_{L6} \\\alpha_{U7} & \alpha_{CU7} & \alpha_{CL7} & \alpha_{L7} \\\alpha_{U8} & \alpha_{CU8} & \alpha_{CL8} & \alpha_{L8}\end{bmatrix}} = {M^{T} \times M}}} & (17)\end{matrix}$

[0068] Further, the column vector in the right side can be expressed asthe following Equation (18) using the interference matrix M in Equation(2) and the errors P₀ before adjustment between the temperaturesdetected by the eight thermocouple-mounted wafers and the targettemperature in Equation (3). $\begin{matrix}{\begin{bmatrix}{\left( {- 1} \right) \times \left( {\left( {\alpha_{U1} \times E_{1}} \right) + \ldots + \left( {\alpha_{U8} \times E_{8}} \right)} \right)} \\{\left( {- 1} \right) \times \left( {\left( {\alpha_{CU1} \times E_{1}} \right) + \ldots + \left( {\alpha_{CU8} \times E_{8}} \right)} \right)} \\{\left( {- 1} \right) \times \left( {\left( {\alpha_{CL1} \times E_{1}} \right) + \ldots + \left( {\alpha_{CL8} \times E_{8}} \right)} \right)} \\{\left( {- 1} \right) \times \left( {\left( {\alpha_{L1} \times E_{1}} \right) + \ldots + \left( {\alpha_{L8} \times E_{8}} \right)} \right)}\end{bmatrix} = {{\left( {- 1} \right) \times \begin{bmatrix}\alpha_{U1} & \alpha_{CU1} & \alpha_{CL1} & \alpha_{L1} \\\alpha_{U2} & \alpha_{CU2} & \alpha_{CL2} & \alpha_{L2} \\\alpha_{U3} & \alpha_{CU3} & \alpha_{CL3} & \alpha_{L3} \\\alpha_{U4} & \alpha_{CU4} & \alpha_{CL4} & \alpha_{L4} \\\alpha_{U5} & \alpha_{CU5} & \alpha_{CL5} & \alpha_{L5} \\\alpha_{U6} & \alpha_{CU6} & \alpha_{CL6} & \alpha_{L6} \\\alpha_{U7} & \alpha_{CU7} & \alpha_{CL7} & \alpha_{L7} \\\alpha_{U8} & \alpha_{CU8} & \alpha_{CL8} & \alpha_{L8}\end{bmatrix}^{T} \times \begin{bmatrix}E_{1} \\E_{2} \\E_{3} \\E_{4} \\E_{5} \\E_{6} \\E_{7} \\E_{8}\end{bmatrix}} = {\left( {- 1} \right) \times M^{T} \times P_{0}}}} & (18)\end{matrix}$

[0069] Consequently, the simultaneous equations of Equation (16) can beexpressed as Equation (19). $\begin{matrix}{{\left\lbrack {M^{T} \times M} \right\rbrack \times \begin{bmatrix}{\Delta \quad T_{U}} \\{\Delta \quad T_{CU}} \\{\Delta \quad T_{CL}} \\{\Delta \quad T_{L}}\end{bmatrix}} = {\left( {- 1} \right) \times M^{T} \times P_{0}}} & (19)\end{matrix}$

[0070] Multiplying both sides by an inverse matrix [M^(T)×M]⁻¹ from theleft results in Equation (20). $\begin{matrix}{\begin{bmatrix}{\Delta \quad T_{U}} \\{\Delta \quad T_{CU}} \\{\Delta \quad T_{CL}} \\{\Delta \quad T_{L}}\end{bmatrix}\quad = {\begin{matrix}\left\lbrack {M^{T} \times M} \right\rbrack^{- 1}\end{matrix} \times \left( {- 1} \right) \times M^{T} \times P_{0}}} & (20)\end{matrix}$

[0071] Since the items M and P₀ in the right side of Equation (20) arenumerical values already obtained as the interference matrix and theerrors between the temperatures detected by the eightthermocouple-mounted wafers and the target temperature which errors arepresent before adjustment, substituting these values for Equation (20)allows the values ΔT_(U), ΔT_(CU), ΔT_(CL), and ΔT_(L) to be determined.

[0072] The thus calculated elements ΔT_(U), ΔT_(CU), ΔT_(CL), and ΔT_(L)are corrective values for the target temperature for the cascadethermocouples which values minimize the evaluation expression J, whichis used to increase the number of products having a quality of a fixedlevel or higher. In the above described example, the method has beensequentially described, which solves the simultaneous equations createdby the partial differentiation in order to determine the elementsΔT_(U), ΔT_(CU), ΔT_(CL), and ΔT_(L) of ΔC, which minimize theevaluation expression J. In actual operations, however, the simultaneousequations need not be solved, the corrective values for the targettemperature for the cascade thermocouples can be determined by obtainingthe interference matrix M in Equation (2) and the errors P₀ between thetemperatures detected by the eight thermocouple-mounted wafers and thetarget temperature, and substituting the values obtained for the aboveEquation (20). Finally, the corrective values thus determined are usedto execute control, and the results of the adjustment are checked.

[0073] With the configuration shown in FIGS. 1 and 2, the correctivevalues ΔT_(U), ΔT_(CU), ΔT_(CL), and ΔT_(L) calculated above are appliedto the target value for the cascade thermocouples for the respectiveheating zones (U, CU, CL, and L zones). Then, the system is controlledso that these target temperatures equal the temperatures detected by thecascade thermocouples. After a sufficient time to stabilize thetemperature has passed, the errors between the temperatures detected bythe eight thermocouple-mounted wafers and the original targettemperature are checked. If the errors fall within an allowable range,the adjustment is completed. If any error exceeds the allowable range,additional adjustment is executed. The procedure of the additionaladjustment is the same as described above. In this case, however, in thefirst adjustment, the errors between the temperatures detected by theeight thermocouple-mounted wafers and the target temperature whicherrors are present before adjustment are obtained as P₀ (Equation (3))as described above, whereas in the additional adjustment, the errorsbetween the temperatures detected by the eight thermocouple-mountedwafers and the target temperature which errors are present upon thecheck of the results of the first adjustment are obtained as P₀.Subsequently, as described above, the value obtained is substituted forEquation (20) to determine the elements ΔT_(U), ΔT_(CU), ΔT_(CL), andΔT_(L), which are further applied to the target temperatures for thecascade thermocouples for the respective heating zones whichtemperatures have been obtained by the correction during the firstadjustment. Then, the system is controlled so that these targettemperatures equal the temperatures detected by the cascadethermocouples, and the results of the readjustment are checked.Normally, one or two or at most three adjustment steps enableappropriate results to be obtained. If, however, the allowable range isstill exceeded, the procedure is preferably restarted from theobtainment of the interference matrix.

[0074] The temperature control method and thermal treatment apparatusdescribed above are used in a semiconductor manufacturing apparatus as amethod of manufacturing a semiconductor.

[0075] For example, diffusion processes as used when the temperaturecontrol method and thermal treatment apparatus are applied to adiffusion apparatus include the following:

[0076] <1> Pyrogenic oxidation.

[0077] This process comprises burning a hydrogen gas using an oxygen gasto generate vapors and introducing the vapors into a reaction chamber tooxidize a wafer or a film accumulated thereon.

[0078] This treatment uses a temperature between 700 and 1,000° C. andan atmospheric pressure. Considering the oxidation speed, a preferabletreatment temperature is between 800 and 1,000° C.

[0079] <2> Dry oxidation

[0080] This process comprises introducing an oxygen gas into thereaction chamber to oxidize a wafer or a film accumulated thereon. Thistreatment uses a temperature between 700 and 1,000° C. and theatmospheric pressure. Considering the oxidation speed, a preferabletreatment temperature is between 800 and 1,000° C.

[0081] <3> Phosphorous diffusion

[0082] This process comprises introducing phosphorous trioxide (POCl₃),an oxygen gas, and a nitrogen gas into the reaction chamber as a carriergas. This treatment uses a temperature between 800 and 1,000° C. and theatmospheric pressure.

[0083] <4> Anneal process

[0084] This process comprises introducing an inactive gas such as anitrogen gas into the reaction chamber. This treatment uses atemperature between 800 and 1,000° C. and the atmospheric pressure.

[0085] For semiconductor devices, the above processes <1> and <2> arewidely used to isolate elements from one another, separate electrode orwiring layers from one another, produce a gate oxide film for a MOSFET,a memory cell capacity portion of a DRAM, or a mask for impuritydiffusion and ion implantation, or inactivate and protect a surface.Further, the above process <3> is used to dope gate electrodes or wireson a polysilicon film or on a resistor or a contact portion. Moreover,the above process <4> is used to move impurities, intended for animpurity layer, to the interior of a crystal. Embodiment 2.

[0086] In Embodiment 1, the corrective values are determined before theactual process, but the corrective values may be determined directly inthe actual process. In Embodiment 2, only the interference matrix M isdetermined before the actual process, and during the treatment of thesubstrate, that is, during the actual process, the errors P₀ between thetemperatures detected by the eight thermocouple-mounted wafers and thetarget temperature are obtained and substituted for Equation (20) todetermine the corrective values for the target temperature for thecascade thermocouples. The corrective values determined are applied tothe target temperature for temperature control. Embodiment 3.

[0087] In Embodiment 2, with thermocouple-mounted wafers, thethermocouple is exposed during the treatment of the substrate, so thatmetal contamination may result. Thus, instead of thethermocouple-mounted wafers, profile thermocouples may be inserted intothe furnace. A profile thermocouple 200 comprises thermocouples 204covered with a ceramic tube 201 made of quartz, SiC, or the like, asshown in FIG. 3, thus reducing metal contamination that may be caused bythe thermocouples 204. A plurality of (eight) thermocouples 204 aresealed in the ceramic tube 201 by a thermocouple sealing section 202,and temperature detection signals therefrom are obtained via a wire 203.The thermocouples 204 are provided at locations (height locations)corresponding to the respective thermocouple-mounted wafers shown inEmbodiment 1. This profile thermocouple 200 is constituted by a singletube and is suitable for insertion into a reaction chamber with arelatively narrow space. Further, the plurality of profile thermocouplesenable temperature to be detected at more locations. Embodiment 4.

[0088] Furthermore, in detecting the errors between the detectedtemperatures and the target temperature during the actual process, thetemperature control may be executed by providing eight cascadethermocouples 304 as a heater thermocouple (heater-controllingthermocouple) 300 as shown in FIG. 4, obtaining the differences betweenthe temperatures detected by these cascade thermocouples and the targettemperature, as the errors P₀, substituting the errors P₀ for Equation(20) to determine corrective values for the target temperature, andapplying the corrective values thus determined to the targettemperature. The heater-controlling thermocouple 300 shown in FIG. 4comprises eight ceramic tubes 301 of quartz or ceramic installed on athermocouple sealing section 305 in parallel and into each of which thecascade thermocouple 304 is sealed. The heater controlling thermocouple300 is adapted to be bent and interposed between the soaking tube (12 inFIG. 1) and the reaction tube (13 in FIG. 1). These cascadethermocouples 304 are separately sealed into the ceramic tube 301,thereby preventing the interference between the cascade thermocouples304. If there is no possibility that such interference may occur, aplurality of thermocouples may be inserted into one ceramic tube.Embodiment 5.

[0089] In each of the above described embodiments, the interferencematrix M is determined before the actual process. If, however,semiconductor devices or the like are thermally treated which require along time to process the substrate and which are not affected by aslight variation of the temperature effected when the interferencematrix is determined, then during the treatment of the substrate, thatis, during the actual process, the interference matrix as well as theerrors and corrective values may be determined to correct the targettemperature. Embodiment 6.

[0090] In the above described embodiments, the vertical apparatus inwhich the heating zones are vertically formed has been described. Thepresent invention, however, is applicable to a sheet-feed apparatus thatprocesses one to several wafers and in which the heating zones areformed in intra-surface directions (i.e., in parallel with a surface) ofthe wafer.

[0091]FIGS. 5A and 5B show this sheet-feed apparatus. FIG. 5A is a sideview and FIG. 5B is a top view showing a plurality of zones. In thiscase, if radiation thermometers are used instead of thethermocouple-mounted wafers, non-contact measurements can be carriedout, thereby easily preventing metal contamination. The sheet-feedapparatus shown in FIGS. 5A and 5B comprises a heating furnace 1 havinga reaction tube 1 a constituting a process chamber, and the heatingfurnace 1 has a susceptor 4 on which a wafer (substrate 2) is placed.The interior of the heating furnace 1 is heated to a predeterminedtemperature, while a reaction gas is supplied to the furnace, to form athin film on the substrate 2. A heater 3 acting as a heating source hasthree zones <1>, <2>, and <3> as heating zones, and heat from each zoneis principally absorbed by the susceptor 4, supporting the substrate 2having a relatively large thermal capacity, and by the substrate 2 and areaction gas flowing from a gas tube 5.

[0092] As control sensors, radiation thermometers (infrared radiationthermometers: second temperature detectors) 6A to 6E are provided sothat each heating zone (<1>, <2>, and <3>) are controlled by two ofthem, and heater thermocouples 7A to 7C are also provided which are usedto monitor or control the heater and which correspond to the cascadethermocouples of the vertical apparatus. A temperature adjuster ADconverting section 8 subjects thermocouple signals to AD conversions,and transmits the resulting temperature digital data to a temperatureadjuster control section 9. The radiation thermometers 6A to 6F alsosubjects sensor signals to AD conversions, and transmits the resultingtemperature digital data to the temperature adjuster control section 9.The temperature adjuster control section 9 uses the received temperaturedigital data to execute control (PID) calculations to thereby determineheat power values (0 to 100%). Target temperatures used for this PIDcontrol are corrected using the already described corrective values.Further, a thyrister controller 410 receives an electric signal (0 to100%) from the temperature adjuster controller 9 to activate the heaterthrough thyrister ignition. Furthermore, connected with the temperatureadjuster control section 9 are a main controller 412 via an alternatingpower supply 411 and a temperature adjuster display operating section413. Moreover, the alternating power supply 414 is connected to theradiation thermometers 6A to 6F and the temperature adjuster controlsection 9.

[0093] The radiation thermometers 6A to 6F according to this embodimentare provided inside the reaction tube 1 a for accurately measuring andcontrolling the temperature of the susceptor 4 as a measurement target.The radiation thermometers are used as temperature sensors because theyenable non-contact measurements and are quick to respond. Further, whenthermocouples are used instead of the radiation thermometers, they mustbe mounted on the susceptor due to their measurement principle, thushampering the susceptor 4 from rotating in this case. As describedabove, the sheet-feed apparatus has zones the number of which isdifferent from that of the vertical apparatus, so that the numbers ofcorresponding heater thermocouples (first temperature detectors) andradiation thermometers (second temperature detectors) are changed asappropriate.

[0094] Even for such a sheet-feed apparatus, the application of thepresent invention makes the temperature of the wafer more uniform in itsintra-surface directions.

[0095] Furthermore, even for the vertical apparatus, by providing aplurality of thermocouples on the same surface of thethermocouple-mounted wafer described in connection with the verticalapparatus, the temperature can be made more uniform in the intra-surfacedirections of the wafer.

[0096] The several embodiments of the methods of controlling thetemperatures of the areas of the treatment targets have been described.By executing these procedures, the temperatures of the areas of thetreatment targets can be promptly and reliably adjusted so to be uniformwithout any skilled operator. Further, by programming the aboveprocedures using a computer and incorporating them in a temperaturecontroller or the like as software, the temperatures of the areas of thetreatment targets can be automatically adjusted so to be uniform withoutthe assistance of a skilled operator. For example, this computerizationenables the automation of the setting of the target temperature for thecascade thermocouples, which produces such results as shown in FIG. 9.In the above example, the eight thermocouple-mounted wafers are used,but more such wafers may be used. Alternatively, the temperature of awider area can be fine-tuned so as to be more uniform by using athermocouple having a mechanism that passes by vertically arrangedwafers from the top to bottom thereof as well as means similar to thosedescribed above. Further, in the above examples, the cascadethermocouples are used for control, but if the cascade thermocouplescannot be always installed, the heater thermocouples can be used insteadto execute similar adjustment. Furthermore, the radiation thermometerscan be used in place of the thermocouple-mounted wafer.

[0097] As described above, according to the present invention, correctedtarget temperatures can be automatically set for a plurality of areasactually subjected to temperature control, using a computer. Therefore,accurate and optimal thermal treatment can be promptly executed even ifthere is not any skilled operator.

What is claimed is:
 1. A temperature control method of controlling a heating apparatus having at least two heating zones so as to adjust temperatures detected at predetermined locations to a target value therefor, said method comprising: detecting temperatures at said predetermined locations the number of which is larger than the number of said heating zones and at least one of which is in each of said heating zones; and controlling said heating apparatus in such a manner that said target temperature falls between a maximum value and a minimum value of a plurality of temperatures detected at a plurality of detected predetermined locations.
 2. The temperature control method according to claim 1, wherein first temperature detectors are disposed at first predetermined locations corresponding to said respective zones, and are used for a temperature control method of controlling said heating apparatus in such a manner that temperatures detected by said first temperature detectors equal a first target temperature, and wherein second temperature detectors are disposed at second predetermined locations which are closer to a treatment target than said first predetermined locations, to obtain an interference matrix M as well as differences P₀ between a second target temperature for said second temperature detectors and temperatures detected by said second temperature detectors, said interference matrix M being a matrix of coefficients indicative of the extents of variations of temperatures detected by said second temperature detectors when said first target temperature for said first temperature detectors is varied, and wherein said first target temperature is corrected on the basis of said interference matrix M and said errors P₀.
 3. The temperature control method according to claim 2, further comprising: determining new errors P₀′ by performing temperature control using said corrected first target temperature; and further correcting said corrected first target temperature using said new errors P₀′ and said interference matrix M.
 4. A temperature control method for controlling an apparatus which includes a process chamber, a heating apparatus having at least one heating zone for heating a treatment target provided in said process chamber, and first temperature detectors provided at least one for each zone for detecting heating temperatures provided by said heating apparatus at first predetermined locations, wherein said heating apparatus is controlled on the basis of first detected temperatures detected by said first temperature detectors and a first target temperature for said first detected temperatures, and wherein a plurality of second temperature detectors are disposed at second predetermined locations the number of which is larger than that of said heating zones and which are closer to said treatment target than said first predetermined locations, said second temperature detectors being operable to detect heating temperatures provided by said heating apparatus, said method comprising: comparing second detected temperatures detected by said second temperature detectors with a second target temperature for the second detected temperatures to obtain corrective values for said first target temperature; and correcting said first target temperature by said corrective values to perform temperature control.
 5. The temperature control method according to claim 4, wherein said corrective values are obtained before an actual process of actually treating a substrate to be treated.
 6. A thermal treatment apparatus, comprising: a process chamber; a heating apparatus having at least two heating zones and being subjected to temperature control in such a manner that temperatures detected at predetermined locations equal a target temperature therefor; a plurality of temperature detectors for detecting temperatures at predetermined locations the number of which is larger than the number of said heating zones and at least one of which is in each of said heating zones; and a control device for controlling said heating apparatus in such a manner that said target temperature falls between a maximum value and a minimum value of a plurality of temperatures detected by means of said plurality of temperature detectors.
 7. A method of manufacturing a semiconductor device, in which a target substrate is subjected to a heating process by controlling a heating apparatus having at least two heating zones in such a manner that temperatures detected at predetermined locations equal a target temperature therefor, said method comprising: detecting temperatures at predetermined locations the number of which is larger than the number of said heating zones and at least one of which is in each of said heating zones; and controlling said heating apparatus in such a manner that said target temperature falls between a maximum value and a minimum value of a plurality of temperatures detected at a plurality of detected predetermined locations. 